Symbolic regression embedding dimensionality analysis

ABSTRACT

A dimensionality analysis method, system, and computer program product, include defining a set of symbols along with a grammatical rule that define a syntax of a valid expression having a valid physical dimensionality relationship.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a Continuation Application of U.S. patentapplication Ser. No. 15/843,118, filed on Dec. 15, 2017, the entirecontents of which are hereby incorporated by reference.

TECHNICAL FIELD

The present invention relates generally to a dimensionality analysismethod, and more particularly, but not by way of limitation, to asystem, method, and recording medium for describing admissiblerelationships between quantities, and incorporating the use of suchgrammar in the automatic discovery of symbolic expressions.

BACKGROUND

Derivation of meaningful symbolic expressions requires consistency withrespect to the attributes of the quantities involved. While expressionscan be consistent in terms of their syntax, they may still beinconsistent with respect to the quantities they comprise (e.g.,dimensions) and the admissible relations between them. For example, whendealing with physical entities, dimensionality analysis (i.e. ensuringthat the unit dimensions of all entities are consistent) is critical toattain meaningful expressions. Conventional techniques includefixed-form solutions but do not consider free-form methodology.Moreover, a very large amount of data is needed to cover the spacebetween points. Such a solution is not generalized. It is difficult tointerpret data in-between points.

SUMMARY

In an exemplary embodiment, the present invention can provide acomputer-implemented dimensionality analysis method, the methodincluding defining a grammar that describes an admissible relationshipbetween quantities in a data set, discovering symbolic expressions thatintrinsically account for a dimensionality analysis based on thegrammar, conducting a search that determines which valid expression inthe data set satisfy the grammar, and selecting the expression that fitsthe dataset and minimizes a measure of a complexity of the expression.

One or more other exemplary embodiments include a computer programproduct and a system.

Other details and embodiments of the invention will be described below,so that the present contribution to the art can be better appreciated.Nonetheless, the invention is not limited in its application to suchdetails, phraseology, terminology, illustrations and/or arrangements setforth in the description or shown in the drawings. Rather, the inventionis capable of embodiments in addition to those described and of beingpracticed and carried out in various ways and should not be regarded aslimiting.

As such, those skilled in the art will appreciate that the conceptionupon which this disclosure is based may readily be utilized as a basisfor the designing of other structures, methods and systems for carryingout the several purposes of the present invention. It is important,therefore, that the claims be regarded as including such equivalentconstructions insofar as they do not depart from the spirit and scope ofthe present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

Aspects of the invention will be better understood from the followingdetailed description of the exemplary embodiments of the invention withreference to the drawings, in which

FIG. 1 exemplarily shows a high-level flow chart for a dimensionalityanalysis method 100;

FIGS. 2A and 2B exemplarily depict a dimensionality analysis accordingto an embodiment of the present invention;

FIG. 3 depicts a cloud computing node 10 according to an embodiment ofthe present invention;

FIG. 4 depicts a cloud computing environment 50 according to anembodiment of the present invention; and

FIG. 5 depicts abstraction model layers according to an embodiment ofthe present invention.

DETAILED DESCRIPTION

The invention will now be described with reference to FIG. 1-5, in whichlike reference numerals refer to like parts throughout. It is emphasizedthat, according to common practice, the various features of the drawingare not necessarily to scale. On the contrary, the dimensions of thevarious features can be arbitrarily expanded or reduced for clarity.

With reference now to the example depicted in FIG. 1, the dimensionalityanalysis method 100 includes various steps to reduce the search spacefor the discovery of symbolic expressions thereby accelerating thesearch and discovering expressions to have interpretations that are moremeaningful (i.e. have correct physical interpretation). As shown in atleast FIG. 2, one or more computers of a computer system 12 according toan embodiment of the present invention can include a memory 28 havinginstructions stored in a storage system to perform the steps of FIG. 1.

Although one or more embodiments (see e.g., FIGS. 3-5) may beimplemented in a cloud environment 50 (see e.g., FIG. 4), it isnonetheless understood that the present invention can be implementedoutside of the cloud environment.

With reference now to FIG. I in step 101, a grammar is defined thatdescribes an admissible relationship between quantities in a data set(i.e., discovery of such a system of unknown form, based on experimentaldata set (observations)). In step 102, symbolic expressions arediscovered that intrinsically account for a dimensionality analysisbased on the grammar (i.e., discovering the functional form of adifferential equation (as well as the relevant parameters)). In step103, a search is conducted that determines which valid expression in thedata set satisfies the grammar, and the expression is selected that fitsthe dataset and minimizes a measure of a complexity of the expression.

That is, the method 100 defines a grammar that describes admissiblerelationships between quantities, and incorporates the use of suchgrammar in the automatic discovery of symbolic expressions (e.g., thatare valid, consistent, etc.) There are several advantages including thatthe search space for the discovery of symbolic expressions is reduced,thereby accelerating the search. Further, the expressions discoveredthis way will be more meaningful (i.e. have correct physicalinterpretation). The method 100 discovers symbolic expressions thatintrinsically account for dimensionality analysis. The grammar preventsoperations between relationships that are not possible (i.e., a quantitywith “m/s” as unit being added to a quantity with joules).

A set of symbols are included that represent viable operators,variables, coefficients, along with grammatical rules that define thesyntax of valid expressions. In addition to the set of symbols, a set ofattributes associated with each variable that specifies its dimensionalproperties are discovered, and a grammar that specifics which operationsbetween variables are allowed, depending on the correspondingattributes. A search is thus conducted that determines which validexpressions satisfy the grammar and chooses the expression that fits thedata by minimizing a measure of complexity (i.e., no specific formbecause differential operator become an extension of the formal languagesuch that there are theoretical guarantees that such simplest equationcan be found and certified). Thus, the invention takes an input,converts the input and issues an output such that the units/dimensionsmatch the conversion.

In another embodiment, the attributes and grammar are generalized toproperties that do not relate to dimensional properties, but encodeabstract relationships between entities.

FIGS. 2A and 2B exemplarily depict a dimensional attribute tree (DAT)having for examples Ranks 1-5 (the invention of course not being limitedto five ranks). Rank I may show the solution ending in seconds whereasrank 4 may show integers and rank 5 may show velocity. Symbolicexpressions can be represented by an expression tree. An expression treeis a way of parsing an expression, thereby ensuring that the expressionfollows the correct syntax. This invention introduces the constructionof an accessory tree, called a “dimensional attribute tree” (DAT) in thefollowing, that automatically encodes the dimensional attributes of eachnode in the expression tree. The DAT has the same topology as theexpression tree. Every operator in a node of the expression tree ismirrored in the corresponding node in the DAT, and applies thecorresponding operation to compute the resulting dimensional attributes(i.e., an equation for spring force and the corresponding units isexpressed). Invalid operations in the DAT can be detected and forbiddenin the expression tree as well (i.e., in rank 4, l and g cannot becombined because meters and meters per second squared cannot becombined).

Thus, symbolic expressions can be represented by the expression treecreated by method 100, which can parse an expression, ensuring that theexpression follows the correct syntax. This method 100 can build the DATas shown in FIGS. 2A and 2B, that automatically encodes the dimensionalattributes of each node in the expression tree. Thus, with theexpression tree and the DAT. Thereby, the search space for the discoveryof symbolic expressions is reduced which accelerates the search and theexpressions discovered this way will be more meaningful.

That is, the method 100 can provide a globally optimal, free-formmethodology for the discovery of mathematical models involvingdifferential equations

Exemplary Aspects, Using a Cloud Computing Environment

Although this detailed description includes an exemplary embodiment ofthe present invention in a cloud computing environment, it is to beunderstood that implementation of the teachings recited herein are notlimited to such a cloud computing environment. Rather, embodiments ofthe present invention are capable of being implemented in conjunctionwith any other type of computing environment now known or laterdeveloped.

Cloud computing is a model of service delivery for enabling convenient,on-demand network access to a shared pool of configurable computingresources (e.g. networks, network bandwidth, servers, processing,memory, storage, applications, virtual machines, and services) that canbe rapidly provisioned and released with minimal management effort orinteraction with a provider of the service. This cloud model may includeat least five characteristics, at least three service models, and atleast four deployment models.

Characteristics are as follows:

On-demand self-service: a cloud consumer can unilaterally provisioncomputing capabilities, such as server time and network storage, asneeded automatically without requiring human interaction with theservice's provider.

Broad network access: capabilities are available over a network andaccessed through standard mechanisms that promote use by heterogeneousthin or thick client platforms (e.g., mobile phones, laptops, and PDAs).

Resource pooling: the provider's computing resources are pooled to servemultiple consumers using a multi-tenant model, with different physicaland virtual resources dynamically assigned and reassigned according todemand. There is a sense of location independence in that the consumergenerally has no control or knowledge over the exact location of theprovided resources but may be able to specify location at a higher levelof abstraction (e.g., country, state, or datacenter).

Rapid elasticity: capabilities can be rapidly and elasticallyprovisioned, in some cases automatically, to quickly scale out andrapidly released to quickly scale in. To the consumer,the capabilitiesavailable for provisioning often appear to be unlimited and can bepurchased in any quantity at any time.

Measured service: cloud systems automatically control and optimizeresource use by leveraging a metering capability at some level ofabstraction appropriate to the type of service (e.g., storage,processing, bandwidth, and active user accounts). Resource usage can bemonitored, controlled, and reported providing transparency for both theprovider and consumer of the utilized service.

Service Models are as follows:

Software as a Service (SaaS): the capability provided to the consumer isto use the provider's applications running on a cloud infrastructure.The applications are accessible from various client circuits through athin client interface such as a web browser (e.g., web-based e-mail).The consumer does not manage or control the underlying cloudinfrastructure including network, servers, operating systems, storage,or even individual application capabilities, with the possible exceptionof limited user-specific application configuration settings.

Platform as a Service (PaaS): the capability provided to the consumer isto deploy onto the cloud infrastructure consumer-created or acquiredapplications created using programming languages and tools supported bythe provider. The consumer does not manage or control the underlyingcloud infrastructure including networks, servers, operating systems, orstorage, but has control over the deployed applications and possiblyapplication hosting environment configurations.

Infrastructure as a Service (IaaS): the capability provided to theconsumer is to provision processing, storage, networks, and otherfundamental computing resources where the consumer is able to deploy andrun arbitrary software, which can include operating systems andapplications. The consumer does not manage or control the underlyingcloud infrastructure but has control over operating systems, storage,deployed applications, and possibly limited control of select networkingcomponents (e.g., host firewalls).

Deployment Models are as follows:

Private cloud: the cloud infrastructure is operated solely for anorganization. It may be managed by the organization or a third party andmay exist on-premises or of premises.

Community cloud: the cloud infrastructure is shared by severalorganizations and supports a specific community that has shared concerns(e.g., mission, security requirements, policy, and complianceconsiderations). It may be managed by the organizations or a third partyand may exist on-premises or off-premises.

Public cloud: the cloud infrastructure is made available to the generalpublic or a large industry group and is owned by an organization sellingcloud services.

Hybrid cloud: the cloud infrastructure is a composition of two or moreclouds (private, community, or public) that remain unique entities butare bound together by standardized or proprietary technology thatenables data and application portability (e.g., cloud bursting forload-balancing between clouds).

A cloud computing environment is service oriented with a focus onstatelessness, low coupling, modularity, and semantic interoperability.At the heart of cloud computing is an infrastructure comprising anetwork of interconnected nodes.

Referring now to FIG. 3, a schematic of an example of a cloud computingnode is shown. Cloud computing node 10 is only one example of a suitablenode and is not intended to suggest any limitation as to the scope ofuse or functionality of embodiments of the invention described herein.Regardless, cloud computing node 10 is capable of being implementedand/or performing any of the functionality set forth herein.

Although cloud computing node 10 is depicted as a computer system/server12 it is understood to be operational with numerous other generalpurpose or special purpose computing system environments orconfigurations. Examples of well-known computing systems, environments,and/or configurations that may be suitable for use with computersystem/server 12 include, but are not limited to, personal computersystems, server computer systems, thin clients, thick clients, hand-heldor laptop circuits, multiprocessor systems, microprocessor-basedsystems, set top boxes, programmable consumer electronics, network PCs,minicomputer systems, mainframe computer systems, and distributed cloudcomputing environments that include any of the above systems orcircuits, and the like.

Computer system/server 12 may be described in the general context ofcomputer system-executable instructions, such as program modules, beingexecuted by a computer system. Generally, program modules may includeroutines, programs, objects, components, logic, data structures, and soon that perform particular tasks or implement particular abstract datatypes. Computer system/server 12 may be practiced in distributed cloudcomputing environments where tasks are performed by remote processingcircuits that are linked through a communications network. In adistributed cloud computing environment, program modules may be locatedin both local and remote computer system storage media including memorystorage circuits.

Referring again to FIG. 3, computer system/server 12 is shown in theform of a general-purpose computing circuit. The components of computersystem/server 12 may include, but are not limited to, one or moreprocessors or processing units 16, a system memory 28, and a bus 18 thatcouples various system components including system memory 28 toprocessor 16.

Bus 18 represents one or more of any of several types of bus structures,including a memory bus or memory controller, a peripheral bus, anaccelerated graphics port, and a processor or local bus using any of avariety of bus architectures. By way of example, and not limitation,such architectures include Industry Standard Architecture (ISA) bus,Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, VideoElectronics Standards Association (VESA) local bus, and PeripheralComponent Interconnects (PCI) bus.

Computer system/server 12 typically includes a variety of computersystem readable media. Such media may be any available media that isaccessible by computer system/server 12, and it includes both volatileand non-volatile media, removable and non-removable media.

System memory 28 can include computer system readable media in the formof volatile memory, such as random access memory (RAM) 30 and/or cachememory 32. Computer system/server 12 may further include otherremovable/non-removable, volatile/non-volatile computer system storagemedia. By way of example only, storage system 34 can be provided forreading from and writing to a non-removable, non-volatile magnetic media(not shown and typically called a “hard drive”). Although not shown, amagnetic disk drive for reading from and writing to a removable,non-volatile magnetic disk (e.g., a “floppy disk”), and an optical diskdrive for reading from or writing to a removable, non-volatile opticaldisk such as a CD-ROM, DVD-ROM or other optical media can be provided.In such instances, each can be connected to bus 18 by one or more datamedia interfaces. As will be further depicted and described below,memory 28 may include at least one program product having a set (e.g.,at least one) of program modules that are configured to carry out thefunctions of embodiments of the invention.

Program/utility 40, having a set (at least one) of program modules 42,may be stored in memory 28 by way of example, and not limitation, aswell as an operating system, one or more application programs, otherprogram modules, and program data. Each of the operating system, one ormore application programs, other program modules, and program data orsome combination thereof, may include an implementation of a networkingenvironment. Program modules 42 generally carry out the functions and/ormethodologies of embodiments of the invention as described herein.

Computer system/server 12 may also communicate with one or more externalcircuits 14 such as a keyboard, a pointing circuit, a display 24, etc.;one or more circuits that enable a user to interact with computersystem/server 12; and/or any circuits (e.g., network card, modem, etc.)that enable computer system/server 12 to communicate with one or moreother computing circuits. Such communication can occur via Input/Output(I/O) interfaces 22. Still yet, computer system/server 12 cancommunicate with one or more networks such as a local area network(LAN), a general wide area network (WAN), and/or a public network (e.g.,the Internet) via network adapter 20. As depicted, network adapter 20communicates with the other components of computer system/server 12 viabus 18. It should be understood that although not shown, other hardwareand/or software components could be used in conjunction with computersystem/server 12. Examples, include, but are not limited to: microcode,circuit drivers, redundant processing units, external disk drive arrays,RAID systems, tape drives, and data archival storage systems, etc.

Referring now to FIG. 4, illustrative cloud computing environment 50 isdepicted. As shown, cloud computing environment 50 comprises one or morecloud computing nodes 10 with which local computing circuits used bycloud consumers, such as, for example, personal digital assistant (PDA)or cellular telephone 54A, desktop computer 54B, laptop computer 54C,and/or automobile computer system 54N may communicate. Nodes 10 maycommunicate with one another. They may be grouped (not shown) physicallyor virtually, in one or more networks, such as Private, Community,Public, or Hybrid clouds as described hereinabove, or a combinationthereof. This allows cloud computing environment 50 to offerinfrastructure, platforms and/or software as services for which a cloudconsumer does not need to maintain resources on a local computingcircuit. It is understood that the types of computing circuits 54A Nshown in FIG. 4 are intended to be illustrative only and that computingnodes 10 and cloud computing environment 50 can communicate with anytype of computerized circuit over any type of network and/or networkaddressable connection (e.g., using a web browser).

Referring now to FIG 5, an exemplary set of functional abstractionlayers provided by cloud computing environment 50 (FIG. 4) is shown. Itshould be understood in advance that the components, layers, andfunctions shown in FIG. 5 are intended to be illustrative only andembodiments of the invention are not limited thereto. As depicted, thefollowing layers and corresponding functions are provided:

Hardware and software layer 60 includes hardware and softwarecomponents. Examples of hardware components include: mainframes 61; RISC(Reduced Instruction Set Computer) architecture based servers 62;servers 63; blade servers 64; storage circuits 65; and networks andnetworking components 66. In some embodiments, software componentsinclude network application server software 67 and database software 68.

Virtualization layer 70 provides an abstraction layer from which thefollowing examples of virtual entities may be provided: virtual servers71; virtual storage 72; virtual networks 73, including virtual privatenetworks; virtual applications and operating systems 74; and virtualclients 75.

In one example, management layer 80 may provide the functions describedbelow. Resource provisioning 81 provides dynamic procurement ofcomputing resources and other resources that are utilized to performtasks within the cloud computing environment. Metering and Pricing 82provide cost tracking as resources are utilized within the cloudcomputing environment, and billing or voicing for consumption of theseresources. In one example, these resources may comprise applicationsoftware licenses. Security provides identity verification for cloudconsumers and tasks, as well as protection for data and other resources.User portal 83 provides access to the cloud computing environment forconsumers and system administrators. Service level management 84provides cloud computing resource allocation and management such thatrequired service levels are met. Service Level Agreement (SLA) planningand fulfillment 85 provide pre-arrangement for, and procurement of,cloud computing resources for which a future requirement is anticipatedin accordance with an SLA.

Workloads layer 90 provides examples of functionality for which thecloud computing environment may be utilized. Examples of workloads andfunctions which may be provided from this layer include: mapping andnavigation 91; software development and lifecycle management 92; virtualclassroom education delivery 93; data analytics processing 4;transaction processing 95; and, more particularly relative to thepresent invention, the Dimensionality analysis method 100.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, configuration data for integrated circuitry, oreither source code or object code written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Smalltalk, C++, or the like, and procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The computer readable program instructions may executeentirely on the user's computer, partly on the user's computer, as astand-alone software package, partly on the user's computer and partlyon a remote computer or entirely on the remote computer or server. Inthe latter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider). In some embodiments, electronic circuitry including,for example, programmable logic circuitry, field-programmable gatearrays (FPGA), or programmable logic arrays (PLA) may execute thecomputer readable program instructions by utilizing state information ofthe computer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

Further, Applicant's intent is to encompass the equivalents of all claimelements, and no amendment to any claim of the present applicationshould be construed as a disclaimer of any interest in or right to anequivalent of any clement or feature of the amended claim.

What is claimed is:
 1. A computer-implemented dimensionality analysismethod, the method comprising: defining a set of symbols along with agrammatical rule that define a syntax of a valid expression having avalid physical dimensionality relationship.
 2. A computer programproduct, the computer program product comprising a non-transitorycomputer-readable storage medium having program instructions embodiedtherewith, the program instructions executable by a computer to causethe computer to perform: defining a set of symbols along with agrammatical rule that define a syntax of a valid expression having avalid physical dimensionality relationship.
 3. A dimensionality analysissystem, comprising: a processor; and a memory, the memory storinginstructions to cause the processor to: define a set of symbols alongwith a grammatical rule that define a syntax of a valid expressionhaving a valid physical dimensionality relationship.